If I'm not mistaken, the category “vector spaces of dimension $\leq n$” (for any $n > 0$) is a counterexample? The zero object, kernels, cokernels, and the normality of kernels can all be computed as they normally are for vector spaces; but products (and hence pullbacks) are missing for obvious reasons of dimension?

The problem seems to be that there's nothing in the definition of “normal” providing a way to build bigger things out of smaller.

On the other hand, I think one can prove “a normal category has pullbacks of monos”; it sounds like that might be what the book is proving here? Maybe that's all that it actually ends up using in the rest of the chapter, and this is just an omission in the statement of the theorem?

Alternatively, if one adds products to the definition of “normal”, then from that together with pullbacks of monos, one can build all pullbacks (the pullback of $f$ and $g$ is the pullback of the appropriate diagonal map (a mono) along $f \times g$).

If I'm not mistaken, the category “vector spaces of dimension $\leq n$” (for any $n > 0$) is a counterexample? The zero object, kernels, cokernels, and the normality of kernels can all be computed as they normally are for vector spaces; but products (and hence pullbacks) are missing for obvious reasons of dimension. [See comments for elaboration.]

The problem is, intuitively, that there's nothing in the definition of “normal” providing a way to build bigger things out of smaller.

On the other hand, I think one can prove “a normal category has pullbacks of monos”; it sounds like that might be what the book is proving here? Maybe that's all that it actually ends up using in the rest of the chapter, and this is just an omission in the statement of the theorem?

Alternatively, if one adds products to the definition of “normal”, then from that together with pullbacks of monos, one can build all pullbacks (the pullback of $f$ and $g$ is the pullback of the appropriate diagonal map (a mono) along $f \times g$).